Given an open set \(U\subset \mathbb {R}^2\)  and a metric space \(X\)  we denote by \(\mathrm {Lip}(U;X)\)  the family of all Lipschitz functions \(\psi :U\rightarrow X.\)  We denote by \(\mathrm {Lip}(\psi )\)  the Lipschitz constant of \(\psi \in \mathrm {Lip}(U;X).\) 
Furthermore, \(GSBD(U;\mathbb {R}^2)\)  denotes the collection of all generalized special functions of bounded deformation (see [1]}, [2]} for details on the definition). Given \(u\in GSBD^2(U;\mathbb {R}^2)\)  we denote by \(e(u(\cdot ))\)  the density of the symmetrized gradient \(Eu=(Du+(Du)^T)/2\)  with respect to Lebesgue measure \(\mathcal {L}^2\)  and by \(J_u\)  the jump set of \(u.\)  Recall that by [2]} \(J_u\)  is \(\mathcal {H}^1\) -rectifiable. We define
\(GSBD^2(U,\mathbb {R}^2):=\lbrace u\in GSBD(U;\mathbb {R}^2):\,\,e(u)\in L^2(U)\rbrace .\) 
